Soft particle clogging in two-dimensional hoppers

We study the outflow of soft particles through quasi-two-dimensional hoppers with both experiments and simulations. The experiments utilize spheres made with hydrogel, silicone rubber, and glass. The hopper chamber has an adjustable exit width and tilt angle (the latter to control the magnitude of gravitational forcing). Our simulation mimics the experiments using purely two-dimensional soft particles with viscous interactions but no friction. Results from both simulations and experiments demonstrate that clogging is easier for reduced gravitational force or stiffer particles. For particles with low or no friction, the average number of particles in a clogging arch depends only on the ratio between hopper exit width and the mean particle diameter. In contrast, for the silicone rubber particles with larger frictional interactions, arches have more particles than the low friction cases. Additionally, an analysis of the number of particles left in the hopper when clogging occurs provides evidence for a hydrostatic pressure effect that is relevant for the clogging of soft particles, but less so for the harder (glass) or frictional (silicone rubber) particles.

Figure 1

Photograph of different types of particles in a clogged state, with the sample chamber fully vertical (maximum force of gravity). (a) Clogging of small hydrogel particles, showing a three-particle arch. The opening width is $w=14.3$ mm $=1.81d$ in terms of the mean particle diameter $d=7.9$ mm. (b) Clogging of glass spheres, showing a four-particle arch with $w=51.5$ mm $=3.32d$, $d=15.5$ mm. (c) Clogging of silicone rubber particles, showing a five-particle arch with $w=30.6$ mm $=2.19d$, $d=14.0$ mm. (d) Clogging of silicone rubber particles, showing a six-particle arch with $w=39.9$ mm $=2.85d$.

Figure 2

Experimental probability of clogging as a function of $w/d$, the ratio of the hopper exit width $w$ to the droplet diameter $d$. Data for different types of particles with the influence of gravity varied by setting different tilt angles. Different symbols represent different types of particles as indicated in the legend. The different colors stand for different tilt angles $\theta$: ${90}^{\circ }$ (dark purple, largest influence of gravity), ${50}^{\circ }$ (dark blue), ${35}^{\circ }$ (light blue), ${20}^{\circ }$ (green), and ${10}^{\circ }$ (red, smallest influence of gravity). The lines are sigmoidal fits to the large hydrogel data and the glass data: $P_{\mathrm{clog}}(w/d)={\{1+exp[(w/d-a)/s]\}}^{-1}$.

Figure 3

Clogging probability from the simulation. (a) Clogging probability as a function of $w/d$ for fixed values of the driving force $g/F_0$. From left to right, $g/F_0={10}^{-2},3\times {10}^{-3},{10}^{-3},3\times {10}^{-4},{10}^{-4}$. The lines are sigmoid fits as in Fig. 2; the fit parameters are plotted in Fig. 4. (b) Clogging probability as a function of $g/F_0$ for fixed values of $w/d$. From left to right, $w/d=2.50,2.25,2.00,1.75,$ and 1.40. The lines are sigmoid fits to the function $P={\{1+exp[(ln(g/F_0)-ln\left(a\right))/s]\}}^{-1}$, with centers $g/F_0=a$ having values $1.1\times {10}^{-3},2.8\times {10}^{-3},6.7\times {10}^{-3},1.2\times {10}^{-2},$ and $1.8\times {10}^{-2}$ from left to right, and widths $s=0.41,0.29,0.28,0.25,$ and 0.19 respectively.

Figure 4

Sigmoidal fit parameters. (a) Centers of sigmoidal fits for different types of particles under varying gravity, with the symbols corresponding to distinct experiments or simulations as indicated in the legend. (b) Width $s$ of sigmoidal fits. The Hong et al. data are from Ref. [19] and are from a different set of hydrogel particles.

Figure 5

(a) Hazard rate: probability of clogging as the next 50 droplets flow out, as a function of the number of particles $N$ left in the hopper. The lines are the best fit to the Gompertz hazard rate (integrated over the next 50 droplets). For $w/d=1.8,1.6,$ and 1.25, the data are taken from simulation runs with 1454, 1697, and 1579 trials respectively. For these data $g/F_0={10}^{-2}$; this matches the blue diamonds in Figs. 3 and 6. (b) Probability of clogging with $N$ particles left in the hopper. The lines are the best fit to the Gompertz distribution. The symbols at the left side of the plot indicate the probability the system does not clog (divided by 50 for comparison to the probability distribution); note that the $w/d=1.25$ data always clog in the simulation (for 1579 runs) so do not have a symbol. The horizontal segments for $N<0$ indicate the expected probability that the system does not clog, based on the Gompertz distribution. The fit parameters are $w/d=1.25$: $h_0=0.040\pm 0.006$, $b=0.0055\pm 0.0003$; $w/d=1.6$: $h_0=0.014\pm 0.001$, $b=0.0067\pm 0.0003$; $w/d=1.8$: $h_0=0.0073\pm 0.0008$, $b=0.0095\pm 0.0007$.

Figure 6

(a) Gompertz distribution fitting parameter $b$ and (b) Gompertz distribution fitting parameter $h_0$, as a function of opening width $w/d$. The data are from the simulation and different symbols and colors correspond to different fixed values of the driving force $g/F_0$. From left to right, $g/F_0={10}^{-2},3\times {10}^{-3},{10}^{-3},3\times {10}^{-4},{10}^{-4}$. In (a), data are not shown for the two smallest values of $g/F_0$ as $b$ is consistent with zero for those data; see the text for a discussion. Representative error bars are drawn for three of the data sets, and represent 90% confidence intervals. The symbols are the same in both panels and match the symbols of Fig. 3. The short vertical lines in (b) indicate the value of $w/d$ for each data set at which $P_{\mathrm{clog}}=1/2$, based on the sigmoid fitting shown in Fig. 3.

Figure 7

(a) Gompertz distribution fitting parameter $b$. The black line is the curve $b=g$. (b) Gompertz distribution fitting parameter $h_0$. The error bars represent 90% confidence intervals. The black line has a slope of $-1$ for comparison. In both panels, the data are from the simulation and are plotted as a function of $g/F_0$, for fixed values of the opening width $w/d$. From left to right, $w/d=2.50,2.25,2.00,1.75,$ and 1.40. The symbols are the same in both panels and match the symbols of Fig. 3. The short vertical bars at the bottom of (b) indicate the value of $g/F_0$ for each data set at which $P_{\mathrm{clog}}=1/2$, based on the sigmoid fitting shown in Fig. 3.

Figure 8

Histograms of probability of clogging with $N$ particles left in the hopper for (a) small hydrogel particles ($w/d=2.03$), (b) silicone rubber particles ($w/d=3.03)$, and (c) glass particles ($w/d=2.91$). All data are taken at tilt angle $\theta ={50}^{\circ }$. The solid black lines are the best fit to the Gompertz distribution; note that this is a reasonable fit in (a) and a less reasonable fit in (b) and (c), showing that the latter data are not consistent with a Gompertz distribution. The symbols at the left side of the plot indicate the probability the system does not clog (divided by 20 for comparison to the probability distribution). The horizontal segments for $N<0$ indicate the expected probability that the system does not clog, based on the Gompertz distribution. The fitting parameters are (a) $h_0=0.011\pm 0.004,b=0.009\pm 0.003$; (b) $h_0=0.007\pm 0.1,b\approx 0$; (c) $h_0=0.006\pm 0.1,b\approx 0$.

Figure 9

Average arch size as a function of $w/d$, the ratio of the hopper exit width $w$ to the droplet diameter $d$. (a) Data from the simulation. Different values of gravitational driving are indicated by the different symbols and colors; from left to right, $g/F_0$ = 0.03, 0.01, 0.003, 0.001, 0.0003, and 0.0001. The symbol shape and color matches those shown in Fig. 3. (b) The line reprises the simulation data from (a). The symbols are the experimental data using different types of particles under the influence of different values of gravity. The symbols are the same as Fig. 4; in particular, the green squares that are outliers correspond to the silicone rubber particles, which have a significantly higher coefficient of static friction.